Second Kind In Terms Research Articles

Coherent pairs of moment functionals of the second kind and associated orthogonal polynomials and Sobolev orthogonal polynomials

Given a pair of quasi-definite moment functionals {v0,v1} we introduce the concept of coherence of the second kind in terms of an algebraic relation that the corresponding sequences of orthogonal polynomials satisfy. We characterize such moment functionals and give some illustrative examples taking into account they are semiclassical of class at most one. The relation between the corresponding monic Jacobi matrices is stated. For a pair of moment functionals satisfying the coherence property of the second kind, a Sobolev inner product is introduced. The connection formulas between the sequence of monic orthogonal polynomials associated with such a Sobolev inner product and the sequence of monic polynomials orthogonal with respect to the moment functional v0 are given.

Closed formulas for special bell polynomials by Stirling numbers and associate Stirling numbers

We derive two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of associate Stirling numbers of the second kind, give an explicit formula for associate Stirling numbers of the second kind in terms of the Stirling numbers of the second kind, and, consequently, present two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of the Stirling numbers of the second kind.

Scattering and radiation of water waves by a submerged rigid disc in a two-layer fluid

Interaction of water waves with a horizontal rigid disc submerged in the lower layer of a two-layer fluid is studied in three dimensions using linear theory. The governing boundary value problem is reduced to a two-dimensional hypersingular integral equation. This integral equation is further reduced to a one-dimensional Fredholm integral equation of the second kind in terms of a newly defined function. The solution to the latter integral equation is used to compute the total scattering cross section and the hydrodynamic force for the scattering problem and the added mass and the damping coefficient for the radiation problem. Haskind relations connecting the solutions of the radiation and the scattering problems are also derived. The effects of variations of the submergence depth of the disc and the depth of the upper layer on different physical quantities are investigated. We observe amplification of the added mass and the damping coefficient, the total scattering cross section and the hydrodynamic force when the disc goes near the interface or when the height of the upper layer decreases. Known results for a horizontal disc submerged in a single-layer fluid of infinite depth are recovered from the present analysis.

A note on type 2 Changhee and Daehee polynomials

In recent years, many authors have studied Changhee and Dae- hee polynomials in connection with many special numbers and polynomials. In this paper, we investigate type 2 Changhee and Daehee numbers and polynomials and give some identities for these numbers and polynomials in relation to type 2 Euler and Bernoulli numbers and polynomials. In addition, we express the central factorial numbers of the second kind in terms of type 2 Bernoulli, type 2 Changhee and type 2 Daehee numbers of negative integral orders.

On multivariate logarithmic polynomials and their properties

In the paper, the author introduces a new notion “multivariate logarithmic polynomial”, establishes two recurrence relations, an explicit formula, and an identity for multivariate logarithmic polynomials by virtue of the Faà di Bruno formula and two identities for the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, some product inequalities, and logarithmic convexity for multivariate logarithmic polynomials by virtue of some properties of completely monotonic functions.

A new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind

We find an explicit formula for computing the Bernoulli numbers of the second kind in terms of the signed Stirling numbers of the first kind.

Some properties of Fibonacci and Chebyshev polynomials

In this paper, we study the properties of Chebyshev polynomials of the first and second kind and those of Fibonacci polynomials and use an elementary method to give Chebyshev polynomials of the first and second kind in terms of Fibonacci polynomials and vice versa. Finally, we get some identities involving the Fibonacci numbers and the Lucas numbers.

An efficient collocation method for a Caputo two-point boundary value problem

A two-point boundary value problem is considered on the interval $$[0,1]$$ , where the leading term in the differential operator is a Caputo fractional-order derivative of order $$2-\delta $$ with $$0<\delta <1$$ . The problem is reformulated as a Volterra integral equation of the second kind in terms of the quantity $$u'(x)-u'(0)$$ , where $$u$$ is the solution of the original problem. A collocation method that uses piecewise polynomials of arbitrary order is developed and analysed for this Volterra problem; then by postprocessing an approximate solution $$u_h$$ of $$u$$ is computed. Error bounds in the maximum norm are proved for $$u-u_h$$ and $$u'-u_h'$$ . Numerical results are presented to demonstrate the sharpness of these bounds.

On a criterion for the invertibility of integral operators of the second kind in the space of summable functions on the semiaxis

To establish a criterion for the invertibility of integral operators of the second kind in terms of the kernel of the integral part of this operator, a special factorization method is applied. We present a transformation reducing the problem of the invertibility of such an operator to that of the inversion of an integral operator of the second kind with contracting integral part.

Periods of second kind differentials of $(n,s)$-curves

For elliptic curves, expressions for the periods of elliptic integrals of the second kind in terms of theta-constants, have been known since the middle of the 19th century. In this paper we consider the problem of generalizing these results to curves of higher genera, in particular to a special class of algebraic curves, the so-called $(n,s)$-curves. It is shown that the representations required can be obtained by the comparison of two equivalent expressions for the projective connection, one due to Fay-Wirtinger and the other from Klein-Weierstrass. As a principle example, we consider the case of the genus two hyperelliptic curve, and a number of new Thomae and Rosenhain-type formulae are obtained. We anticipate that our analysis for the genus two curve can be extended to higher genera hyperelliptic curves, as well as to other classes of $(n,s)$ non-hyperelliptic curves.

Frictional contact of anisotropic piezoelectric materials indented by flat and semi-parabolic stamps

An exact analysis of frictional contact of anisotropic piezoelectric materials indented by a rigid flat or semi-parabolic stamp is conducted. Fundamental solutions that can lead to the real values of physical quantities are detailed for each eigenvalue distribution. The complicated mixed boundary value problems are reduced into a singular integral equation of the second kind in terms of the unknown surface contact stress beneath the stamp. Employing excellent properties of Jacobi Polynomials, the exact solution of the reduced second kind singular integral equation can be obtained. Exact and explicit expressions of various surface stresses and electric displacement are given in terms of elementary functions. Relationships between the applied load and contact area are derived, and stress intensity factors at stamp edges are given. Numerical results are presented to show the effects of the friction coefficient on various surface stresses and electric displacement. The present investigation could provide a scientific basis for the theoretical and experimental test of contact behaviors of anisotropic piezoelectric materials.

Optimal combinations bounds of root-square and arithmetic means for Toader mean

We find the greatest value α1 and α2, and the least values β1 and β2, such that the double inequalities α1S(a,b) + (1 − α1) A(a,b) 0 with a ≠ b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, S(a,b) = [(a2 + b2)/2]1/2, A(a,b) = (a + b)/2, and \(T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta\) denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.

Two kinds of first-person-oriented content

In this paper, I will argue that two kinds of first-person-oriented content are distinguished in more ways than usually thought and I propose an account that will shed new light on the distinction. The first kind consists of contents of attitudes de se (in a broad sense); the second kind consists of contents that give rise to intuitions of relative truth. I will present new data concerning the two kinds of first-person-oriented content, together with a novel account of propositional content in general, namely based on the notion of an attitudinal object. That notion solves two major problems with Lewis’s account of contents of attitudes de se and clarifies the difference between contents of attitudes de se and contents that give rise to intuitions of relative truth. I will propose an analysis of contents of the second kind in terms of what I call first-person-based genericity, a form of genericity most explicitly expressed by sentences with generic one. I show how the overall account explains the particular semantic properties of sentences giving rise to intuitions of relative truth that distinguish them from sentences with expressions interpreted de se. I will start by introducing Lewis’s account of attitudes de se and the problems that go along with that account. Introducing the notion of an attitudinal object, I will extend the account by an account of the truth conditions of the content of attitudes de se. I then discuss the second kind of first-person-oriented content, which is associated with intuitions of relative truth, and give an account of such contents on the basis of an analysis of generic one. Again making use of attitudinal objects, I will make clear what exactly distinguishes those contents from first-person-oriented contents of the first sort.

The Fourier series expansions of the Legendre incomplete elliptic integrals of the first and second kind

Recently, the Fourier series expansions of the Legendre incomplete elliptic integrals F(φ, k) and E(φ, k) of the first and second kind in terms of the amplitude φ were investigated and found in a series of papers. The expansions were derived in several ways, for instance, by using a hypergeometric series approach, and have coefficients involving either the hypergeometric function or the associated Legendre functions of the second kind. In this paper, it is shown that the Fourier series expansions of F(φ, k) and E(φ, k) can be obtained without any difficulty by applying the usual and more familiar Fourier-series technique. Moreover, as an interesting consequence of this approach, both the recently found expansions and the new expansions with coefficients which are solely linear combinations of the complete elliptic integrals of the first and second kind, K(k) and E(k), are obtained in a unified manner. Furthermore, unlike the previously known, the newly established results make it possible to easily compute the Fourier coefficients of F(φ, k) and E(φ, k) analytically.

Stirling Functions of the Second Kind in the Setting of Difference and Fractional Calculus

The purpose of this article and companion ones is to present a new approach to generalizations of Stirling numbers of the first and the second kind in terms of fractional calculus analysis by using differences and differentiation operators of fractional order. Such an approach allows us to extend the classical Stirling numbers of the first and the second kind in a natural way not only to any positive, but also to any negative order. Moreover, an application of the fractional approach gives us the opportunity to extend the classical Stirling numbers to more general complex functions. In the present article we extend the classical Stirling numbers of the second kind, S(n, k), for the first parameter from a nonnegative integer number n to any complex α. Such constructions, S(α, k), will be defined for any complex α and by when k >0, while S(0, 0) = 1 and when k = 0. We show that S(α, k) with positive α can be represented by the Liouville and Marchaud fractional derivatives of the exponential functions, while for negative α it can be interpreted in terms of Liouville fractional integrals. Many of the main properties of the above Stirling functions are established; they generalize those, well known, for the classical Stirling numbers S(n, k) (). Several new applications are presented. Thus, the sum , for any complex and , is represented as a finite sum involving the S(α + 1, k). Whereas the case α = n is a classical result, even the particular case α = −n gives a new application. Further, Hadamard-type fractional integrals and derivatives, basic in Mellin transform theory on (0, ∞), are represented in terms of infinite series involving the S(α, k) and the powers of the operator . Its corresponding classical discrete version, involving the S(n, k), plays an important role in computational mathematics, combinatorial analysis and discrete mathematics.

A $p,q$-analogue of a Formula of Frobenius

Garsia and Remmel (JCT. A 41 (1986), 246-275) used rook configurations to give a combinatorial interpretation to the $q$-analogue of a formula of Frobenius relating the Stirling numbers of the second kind to the Eulerian polynomials. Later, Remmel and Wachs defined generalized $p,q$-Stirling numbers of the first and second kind in terms of rook placements. Additionally, they extended their definition to give a $p,q$-analogue of rook numbers for arbitrary Ferrers boards. In this paper, we use Remmel and Wach's definition and an extension of Garsia and Remmel's proof to give a combinatorial interpretation to a $p,q$-analogue of a formula of Frobenius relating the $p,q$-Stirling numbers of the second kind to the trivariate distribution of the descent number, major index, and comajor index over $S_n$. We further define a $p,q$-analogue of the hit numbers, and show analytically that for Ferrers boards, the $p,q$-hit numbers are polynomials in $(p,q)$ with nonnegative coefficients.

Comment on Global/Local Method for Low‐Velocity Impact Problems

A revision to a previously developed method for solving the dynamic‐contact problem of a rigid, smooth striker impacting an elastically supported beam is presented. The revision is such that only a local‐static solution is superposed on an elementary‐beam‐theory solution that incorporates the dynamic effects. The local‐static solution is expressed in terms of the difference between an elastic‐finite‐layer solution and a static‐beam‐theory solution. The matching of boundary conditions leads to a Volterra‐integral equation of the second kind in terms of the pressure distribution and contact length as functions of time. These quantities are obtained numerically using a technique developed for the solution of non‐Hertzian‐contact problems. The revised method is shown to more accurately model the low velocity impact response of finite beams through a comparison of experimental results and theoretical predictions.

The periodic orbits of the second kind in terms of Giacaglia's variables with oblateness

Utilizing secular perturbing potential due to oblateness, the existence of periodic orbits of the second kind is established through analytic continuation using Giacaglia's canonical variables in the planar restricted three-body problem when the more massive primary is an oblate spheroid with its equatorial plane coincident with the plane of motion.

Integral Representations for Products of Lamé Functions by Use of Fundamental Solutions

In this paper we present integral representations for products of Lamé functions based on the theory of fundamental solutions. The kernels of these representations involve Legendre functions of the second kind. In particular, we generalize and improve integral representations for external ellipsoidal harmonics mentioned by Erdélyi, Magnus, Oberhettinger and Tricomi [Higher Transcendental Functions III, McGraw Hill, New York, 1955] and for Lamé functions of the second kind in terms of Lamé polynomials studied by Shall [SIAM J. Math. Anal., 11 (1980), pp. 702–723].